7. Lecture Notes on the Ellipsoid Algorithm

The simplex algorithm was the first algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. The Ellipsoid algorithm is the first polynomial-time algorithm discovered for linear programming. The Ellipsoid algorithm was proposed by the Russian mathematician Shor in 1977 for general convex optimization problems, and applied to linear programming by Khachyan in 1979. Contrary to the simplex algorithm, the ellipsoid algorithm is not very fast in practice; however, its theoretical polynomiality has important consequences for combinatorial optimization problems as we will see. Another polynomial-time algorithm, or family of algorithms to be more precise, for linear programming is the class of interior-point algorithms that started with Karmarkar’s algorithm in 1984; interior-point algorithms are also quite practical but they do not have the same important implications to combinatorial optimization as the ellipsoid algorithm does. The problem being considered by the ellipsoid algorithm is: